The Hurwitz quaternion order is a specific order in a quaternion algebra

Quaternion algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...

 over a suitable number field. The order is of particular importance in Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

 theory, in connection with surfaces with maximal symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

, namely the Hurwitz surface

Hurwitz surface

In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with preciselyautomorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms...

s. The Hurwitz quaternion order was studied in 1967 by Goro Shimura

Goro Shimura

is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...

, but first explicitly described by Noam Elkies

Noam Elkies

Noam David Elkies is an American mathematician and chess master.At age 14, Elkies received a gold medal with a perfect score at the International Mathematical Olympiad, the youngest ever to do so...

 in 1998. For an alternative use of the term, see Integer quaternion (both usages are current in the literature).

Definition

Let be the maximal real subfield of where is a 7th-primitive root of unity

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

.
The ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

 of is , where the element can be identified with the positive real . Let be the quaternion algebra

Quaternion algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...

, or symbol algebra


so that and in Also let and . Let


Then is a maximal order of , described explicitly by Noam Elkies

Noam Elkies

Noam David Elkies is an American mathematician and chess master.At age 14, Elkies received a gold medal with a perfect score at the International Mathematical Olympiad, the youngest ever to do so...

.

Module structure

The order is also generated by elements


and


In fact, the order is a free -module over
the basis . Here the generators satisfy the relations


which descend to the appropriate relations in the (2,3,7) triangle group

(2,3,7) triangle group

In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.A note on terminology – the "...

, after quotienting by the center.

Principal congruence subgroups

The principal congruence subgroup defined by an ideal is by definition the group
mod

namely, the group of elements of reduced norm 1 in equivalent to 1 modulo the ideal . The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to PSL(2,R).

Application

The order was used by Katz, Schaps, and Vishne to construct a family of Hurwitz surfaces satsifying an asymptotic lower bound for the systole: where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak

Peter Sarnak

Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...

; see systoles of surfaces

Systoles of surfaces

In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 . Given a closed surface, its systole, denoted sys, is defined to the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to...

.

See also

• (2,3,7) triangle group
  (2,3,7) triangle group
  In the theory of Riemann surfaces and hyperbolic geometry, the triangle group is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84, of its automorphism group.A note on terminology – the "...
  
  
• Klein quartic
  Klein quartic
  In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
  
  
• Macbeath surface
  Macbeath surface
  In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface....
  
  
• First Hurwitz triplet
  First Hurwitz triplet
  In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 . The explanation for this phenomenon is arithmetic...